?
Journalof the AmericanStatisticalAssociation
March 1970, Volume 65, Number329
Theoryand Methods Section
Between
On Birnbaum's
Theoremon theRelatioln
and Likelihood
Conditionality
Suffi:ciency,
J. DURBIN*
variableis requiredto dependonlyon the value of the miniIf the conditioning
statistic,Birnbaum'sprooffails.
mal sufficient
and
Birnbaum's [I] demonstrationthat plausible principlesof sufficiency
conditionalityimplythe likelihoodprinciplehas attracted considerableinterprincipleimpliesthat,as a funcest. However,althoughBirnbaum'ssufficiency
tion of the observations,evidential meaning depends only on the minimal
sufficient
statistic,wherethisexists,the domain of applicationofhis conditionIt
ality principleis not restrictedto statisticswhich are minimallysufficient.
turnsout that if the conditioningvariable is requiredto be part of the minimal
sufficient
statistic,Birnbaum'sprooffails. Consequently,if it is stipulatedthat
of the resultsshould depend on the observations
any analysis or interpretation
statistic,Birnbaum's theorem
onlythroughthe value of the minimalsufficient
is inapplicable.
Birnbaum's startingpoint is the conceptof evidentialmeaningwhich is introducedby him in the followingterms:
of the symbol
"The presentanalysisofthe firstproblembeginswiththe introduction
Ev(E, x) to denotethe evidentialmeaningof a specifiedinstance(E, x) of statistical
evidence; that is, Ev(E, x) stands forthe essentialproperties(whichremainto be
clarified)of the statisticalevidence,as such,providedby the observedoutcomex of
E" [1, p. 270].
the specifiedexperiment
He goes on to definethe concept of a mixtureexperimentas follows:
withcomponents
E is called a mixture(or a mixtureexperiment),
"An experiment
IEh],
if it is mathematicallyequivalent (underrelabelingof sample points) to a two-stage
experimentof the followingform:
(a) An observationh is taken on a randomvariable H having a fixedand known
G. (G does not dependon unknownparametervalues.)
distribution
componentexperiment
(b) The corresponding
Eh is carriedout, yieldingan outcome
Xh" [1, p. 279].
The theoryis based on the followingtwo axioms denoted by Birnbaum as
(S) and (C):
with sample space [x}, and
(S): Let E be any experiment,
"Principleof Sufficiency
statistic (not necessarilyreal-valued). Let E' denote the
let t(x) be any sufficient
derivedexperiment,
havingthe same parameterspace,such thatwhenanyoutcomex
outcomet= t(x) of E' is observed.Then foreach
of E is observedthe corresponding
x, Ev(E, x) =Ev(E', t), wheret=t (x)" [1,p. 2781.
E is (mathematically
equivalentto)
(C): If an experiment
"Principleofconditionality
a mixtureG of componentstEAh,withpossibleoutcomes(EA,xA), then
* J. Durbinis professor
ofstatistics,London School ofEconomics,UniversityofLondon. His publishedworkis
timeseriesand samplesurveys.The authorwishesto thankP. Armitage,A. Birnmainlyin the fieldsofregression,
and A. Stuartforhelpfulcommentson an earlierdraftofthisnote.
baum,V. P. Godambe,L. J. Savage, H. Scheffd,
395
Journalof the AmericanStatisticalAssociation,March 1970
396
Ev(EE,
XA))=Ev(EA, Xh).
That is, the evidential meaning of any outcome (EA, Xh) of any experimentE
havinga mixturestructureis the same as: the evidentialmeaningof the corresponldingoutcomeXAofthe corresponding
componentexperiment
EA, ignoringotherwisethe
E" [1, p. 279].
over-allstructureofthe originalexperiment
Birnbaum'stheorem,derivedas Lemma 2 [1, p. 284], statesthat (S) and (C)
imply,and are impliedby, the likelihoodprinciple,which states that the evidential meaningof an observedoutcomex is characterisedcompletelyby the
likelihoodfunctioncalculated fromx [1, p. 283].
The firstpoint to note is that (S) impliesthat Ev(E, x) depends only on the
minimalsufficient
statistic,wherethis exists.To show this,let t(x) denote the
sufficient
statisticreferredto in (S) and let t'(x) denote a minimalsufficient
statistic.Let E" denote the derived experiment,having the same parameter
space as E', such that whenthe outcomet(x) of E' is observedthe corresponding outcome t'(x) of E" is observed. By (S), Ev(E, x) =-Ev(E', t(x)) =Ev(E",
t'(x)). It followsthat, as a functionof the outcomex, Ev(E, x) dependsonlyon
the minimalsufficient
statistict'(x). (We may speak of theminimalsufficient
statisticis a functionof every
statisticwithoutloss sinceany minimalsufficient
otherminimalsufficient
statistic.)This impliesthat if t'(x) is preservedthe remainingrecordoftheresultsoftheexperimentcan be destroyedwithoutin any
way affectingthe determinationof Ev(E, x).
statisticit
Since evidentialmeaningdepends only on the minimalsufficient
of the rewould seem reasonableto requirethat any analysisor interpretation
sults of the experimentshould depend onlyon the value of the minimalsuffithat the domain of applicientstatistic.This leads naturallyto therequirement
cability of (C) should be restrictedto componentsof the minimal sufficient
led to the followingmodifiedformof (C).
statistic.We are therefore
Modifiedprincipleof conditionality
(C'): If an experimentE is (mathematically equivalent to) a mixtureG of componentsEh, with possible outcomes
statistic,
(Eh, Xh), whereh depends only on the value of the minimalsufficient
then
Ev(E,
(Eh, Xh)) = Ev(Eh, Xh).
ofBirnbaum'sterminology
thisseemsto be consistent
Withinthe framework
withwhat most statisticiansmean by conditionalinference.We now show that
Birnbaum's prooffailswhen (C) is replaced by (C').
Let E be a mixtureexperimentas definedby Birnbaum and considertwo
outcomes,h = i, x = xi and h=j, x = xj wherethe conditionaldensitiesof xi, xj
given i and j are fi(x, 0) and fj(x, 0). Suppose that xi and xj lead to the same
likelihoodfunction,i.e.,fi(xi, 0) = cfj(xj, 0) forall 0 wherec may dependon xi and
xj but not on 0. Denote the densityof the knowndistributionG of h by gh.The
conditionalprobabilitythat h=i and x=xi given that h=i, x=xi or h=j,
x=Xj
is
gifi(xi,0)
gic
gifi(xi,0) + gjfj(xj,0)
gic + gj
On Birnbaum'sTheoremon the LikelihoodPrinciple
397
which does not depend on 0. Consequently,given that h = i, x = xi or h=j,
x = x1,the actual value of h (i or j) caniiot be part of the minimalsufficient
statistic.Thus the statement
Ev(E, (Eh,
Xh))
=
Ev(Eh, Xh),
(1)
whichfollowsfrom(C), does not followfrom(C').
The proofof Birnbaum's theoremdepends cruciallyon the use of (1) in a
situationof the kindwe have been considering.Birnbaumconsiderstwo experimentsE and E' with outcomesx, y having densitiesf(x, 0), g(y, 0) on sample
spaces S, S'. He takes a particularpair of outcomes x', y', which lead to the
same likelihoodfunction,i.e.,f(x', 0) = cg(y',0) forall 0 wherec is a positiveconstantand introducesa (hypothetical)mixtureexperimentE* whosecomponents
are just E and E', taken with equal probabilities.Birnbaum then invokes (C)
to justifythe assertions
Ev(E*, (E, x)) = Ev(E, x) foreach x
E
S, and
Ev(E*, (E', y)) = Ev(E', y) foreach y E S'
((5.1) in [1]). On applying (2) to the cases x=x', y=y' and using (S) his
theoremfollows.
In order to examine the situationfromthe standpointof (C'), let a be an
indicatorvariable such that a = 0 ifE is taken and a = 1 ifE' is taken. The outcome of E* can thenbe writtenas (0, x) or (1, y) accordingas E or E' is taken.
The conditionalprobabilityof (0, x') given { (0, x') or (1, y') } = 1/2f(x', 0)
/ 1/2f(x',0) + 1/2g(y',0) } = c/(I +c) which is independentof 0. Thus for the
particularoutcomes x' and y', the value of a cannot be part of the minimal
sufficient
statistic.Under (C') thereis thereforeno justificationfor the assertions(2) forthe outcomesx = x' or y = y'. Since theseassertionsare essentialfor
Birnbaum'sproofhis theoremfailswhen (C) is replacedby (C').
It must not be thoughtthat the above discussioncan be dismissedas mere
technicalquibbling.On the contrary,I believe that the substitutionof (C') for
(C) involves a serious questioningof Birnbaum's approach to conditionality
It should be noted that his theorem
and of the way he relatesit to sufficiency.
cannot be "disproved"in the ordinarysense owingto the fact that Ev(E, x) is
not defined.If Ev(E, x) is interpretedas likelihoodfromthe outset then the
theoremis triviallytrue.It is forthisreason that the approach of thisnote has
been to indicate the point at which his argumentbreaks down under (C')
ratherthan to attemptto presentcounterexamples.
I wishto emphasizethat my main purposehas not been to argue the case for
the adoption of (C') as a basic principlebut rather,grantedits plausibility,to
demonstratethe consequencesof (C') forBirnbaum'stheoryof conditionality.
Some of the pointsmade in thisnote are criticizedby Savage [4] in the note
immediatelyfollowingit. OtheraspectsofBirnbaum'stheoremare discussedby
{
Fraser[2] and Hajek [3].
REFERENCES
[1] Birnbaum,A., "On the Foundationsof StatisticalInference,"Journalof theAmerican
StatisticalAssociation,57 (1962), 269-306.
398
Journalof the AmericanStatisticalAssociation,March 1970
and LikelihoodPrinciples,"Journalof theAmeri[2] Fraser,D. A. S., "On the Sufficiency
can StatisticalAssociation,58 (1963), 641-7.
[3] Hijek, J., "On Basic Concepts of Statistics,"Proceedingsof theFifthBerkeleySymposium,Vol. I (1967), 139-62.
Journal
[4] Savage, LeonardJ.,"Commentson a WeakenedPrincipleof Conditionality,"
oftheAmericanStatisticalAssociation,65 (1970).